Answer: The correct option is (C) Graph Y.
Step-by-step explanation: The given function is:
[tex]f(x)=\sqrt{x-1}.[/tex]
We are to select the correct graph that corresponds to the above function f(x).
Since we are dealing with real functions, so the function f(x) will be well-defined if the quantity under the square root is greater than or equal to 0.
That is,
[tex]\sqrt{x-1}\geq 0\\\\\Rightarrow x-1\geq 0\\\\\Rightarrow x\geq 1.[/tex]
Therefore, the domain of the function f(x) is [1, ∞).
Option (A)
Here, the domain of the function is (-∞, -1] ≠ [1, ∞).
So, this option is NOT CORRECT.
Option (B)
Here, the domain of the function is [-1, ∞) ≠ [1, ∞).
So, this option is NOT CORRECT.
Option (C)
Here, the domain of the function is [1, ∞), which is the domain of the given function.
So, this option may be CORRECT.
Option (D)
Here, the domain of the function is (-∞, 1] ≠ [1, ∞).
So, this option is NOT CORRECT.
Since the domains of graphs represented by options (A), (B) and (D) do not match with the domains of the given function.
So, options (A), (B) and (D) are NOT CORRECT.
Also, the domain of the graph in option (C) matches with the domain of the given function f(x).
The value of f(x) at x = 1 is
[tex]f(1)=\sqrt{1-1}=0,[/tex]
which is the value of graph Y at x = 0.
And the value of f(x) at x = 2 is
[tex]f(2)=\sqrt{2-1}=1,[/tex]
which is the value of graph Y at x = 1.
So, the graph of the given function f(x) is Y.
Thus, (C) is the correct option.
